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More evidence of the intrinsic beauty of maths, this time shown in a lovely slideshow from the BBC, narrated by mathematician Lasse Rempe from the University of Liverpool.

Rempe works in the area of dynamical systems: systems that change over time and can be found everywhere from the stockmarket to the weather. In the slideshow, he explains how dynamical systems can be generated from very simple polynomials yet produce extremely complex behaviour, and how these systems can be graphically represented by such beautiful images.

As John Barrow told us in our first image-enhanced Plus podcast Cosmic Imagery, mathematical images such as these have actually been responsible for changing science and how we see the world around us . So sit back with your coffee and enjoy the shows!

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...and yesterday was brought to you by the number 54, thanks to the Mathematical Association of America's NumberADay blog. Every working day they post a number and a biography of its interesting properties.

Today's number 11,185,272 is the number of decimal digits in the 46th known Mersenne prime, discovered on Sept. 6, 2008 (you can read more in Prime record broken? on Plus).

54 might seems less significant, but in fact thanks to the MAA Plus now knows that it is the smallest number that can be written as the sum of 3 squares in 3 ways, the number of colored squares on a Rubik’s cube, and is a nonadecagonal (19-gonal) number!

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It's lunchtime, and I'm waiting in the cafe queue to buy a sandwich along with everyone else. What a lot of sandwiches... Imagine if everyone piled their sandwiches one on top of another... I wonder how high the mighty tower of sandwiches might be? Let's see... 60 million people in the UK, say 1 in 8 is having a sandwich right now, each sandwich might be about 3cm thick including filing, so
that's... over 200km high!

You might say I'm thinking too much about sandwiches, but I'm actually exercising my number sense. Which, along with allowing me to make quick guesses about how big things are or how many there might be, also might be helping me get better at calculus and algebra. Researchers Michèle Mazzocco, Lisa Feigenson and Justin Halberda, from John Hopkins University, have shown that being good at
formal mathematics is linked to having a good innate number sense. See, I'm not waiting in the sandwich queue, I'm studying!

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As sporting glories continue in Beijing with the Paralympics taking up where the Olympics left off, many of us have marvelled at the architecture almost as much as at the sporting achievements. One of the Olympic venues, the National Aquatic Centre or Water Cube, seems to be sliced from a giant foam of bubbles, and it turns out mathematics is responsible for this amazing structure.

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You can come out now, it's safe...

Well it's official, the first beam in the Large Hadron Collider at CERN has safely made its way around the 27km tunnel at around 1030 this morning, local time. It was a historic moment, the culmination of over 20 years' work building the biggest experiment the world has seen, and one that many hope will give us a glimpse into the beginnings of
the universe and give experimental evidence to long-held theories fundamental to physics.

"It’s a fantastic moment,” said LHC project leader Lyn Evans, “we can now look forward to a new era of understanding about the origins and evolution of the universe.”

Starting up a major new particle accelarator takes much more than just flipping a switch. Thousands of individual elements have to work in harmony and timings have to be synchronized to under a billionth of a second. The second beam was fired at around 2pm local time, and is now making its way around in the opposite direction. Over the next few weeks, as the people at the LHC learn how to
drive their new toy, they will steer the two beams, finer than a human hair, into a head-on collision. It will be these collisions that will allow the research programme to begin properly.

Once colliding beams have been established, there will be a period of measurement and calibration for the LHC’s four major experiments, and new results could start to appear in about a year's time. Experiments at the LHC will allow physicists to complete a journey that started with Newton's description of gravity. Gravity acts on mass, but so far science is unable to explain the mechanism that
generates mass. Experiments at the LHC will provide the answer. LHC experiments will also try to probe the mysterious dark matter of the universe – visible matter seems to account for just 4% of what must exist, while about a quarter is believed to be dark matter. They will investigate the reason for nature's preference for matter over antimatter, and they will probe matter as it existed at the
very beginning of time.

“The LHC is a discovery machine,” said CERN Director General Robert Aymar, “its research programme has the potential to change our view of the Universe profoundly, continuing a tradition of human curiosity that’s as old as mankind itself.”

You can read more about the LHC and the science it is exploring on Plus

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Prime record broken?

Volunteers have claimed to have found the largest prime number yet — twice within a fortnight! The two new record breakers are both Mersenne primes: numbers which can be written in the form 2p-1, where p is also prime.

Every whole number can be written as a product of prime numbers in a unique way, and this is why the primes are regarded as the building blocks of number theory. Mathematicians have known since antiquity that there are infinitely many primes, but there isn't a formula which describes them all. To check if a number is prime, you have to go through painstaking algorithms that take up a huge
amount of computing power. The task becomes easier when the number you're checking for primeness is a Mersenne number of the form described above.

But still, one computer isn't enough to do the job: the eleven previous largest prime discoverers have all been part of the Great Internet Mersenne Prime Search (GIMPS), which uses the computing power "donated" by tens of thousands of volunteers to chomp through the necessary calculations. The previous record prime — found in September 2006 — would have
taken an ordinary PC 4000 years to find, but with the help of a 70,000 strong computer network, able to perform 22 trillion calculations per second, popped out in "only" nine months.

Why would anyone want to find the largest prime to date? For the fun of it, of course, in true nerdy-style, but there's the added bonus of a $50,000 prize for the first to discover a prime with 10 million digits. On a less frivolous level, primes are extremely useful in cryptography: because factorising large numbers into their prime factors is so
computationally expensive, these factors can, and do, serve as almost unbreakable keys to encrypted messages — like the ones we send over the Internet every time we use our credit cards or send encrypted emails.

Experts are now performing independent checks to verify that the two new numbers really are prime, and are due to report back soon.

To find out more about GIMPS, previous Mersenne prime discoveries, and the role of primes in cryptography, read the Plus articles